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Probabilistic Analysis of Multiparameter Persistence Decompositions into Intervals

Authors Ángel Javier Alonso , Michael Kerber , Primoz Skraba

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ACM Subject Classification
  • Mathematics of computing → Topology
Keywords
  • Topological Data Analysis
  • Multi-Parameter Persistence
  • Decomposition of persistence modules
  • Poisson point processes

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Abstract

Multiparameter persistence modules can be uniquely decomposed into indecomposable summands. Among these indecomposables, intervals stand out for their simplicity, making them preferable for their ease of interpretation in practical applications and their computational efficiency. Empirical observations indicate that modules that decompose into only intervals are rare. To support this observation, we show that for numerous common multiparameter constructions, such as density- or degree-Rips bifiltrations, and across a general category of point samples, the probability of the homology-induced persistence module decomposing into intervals goes to zero as the sample size goes to infinity.

Cite As Get BibTex

Ángel Javier Alonso, Michael Kerber, and Primoz Skraba. Probabilistic Analysis of Multiparameter Persistence Decompositions into Intervals. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 6:1-6:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.SoCG.2024.6

Author Details

Ángel Javier Alonso
  • Institute of Geometry, Graz University of Technology, Austria
Michael Kerber
  • Institute of Geometry, Graz University of Technology, Austria
Primoz Skraba
  • Queen Mary University of London, United Kingdom

Funding

  • Alonso, Ángel Javier: Austrian Science Fund (FWF) grant P 33765-N.
  • Kerber, Michael: Austrian Science Fund (FWF) grant P 33765-N.

Acknowledgements

The authors thank Jan Jendrysiak for helpful discussions. M.K. and P.S. acknowledge the Dagstuhl Seminar 23192 “Topological Data Analysis and Applications” that initiated this collaboration.

References

  1. Ángel Javier Alonso and Michael Kerber. Decomposition of zero-dimensional persistence modules via rooted subsets. In 39th International Symposium on Computational Geometry, SoCG 2023, 2023. URL: https://doi.org/10.4230/LIPIcs.SoCG.2023.7.
  2. Ángel Javier Alonso, Michael Kerber, Tung Lam, and Michael Lesnick. Delaunay bifiltrations of functions on point clouds. In Symposium on Discrete Algorithms SODA 2024, 2024. URL: https://doi.org/10.1137/1.9781611977912.173.
  3. Ángel Javier Alonso, Michael Kerber, and Siddharth Pritam. Filtration-domination in bifiltered graphs. In Proceedings of the Symposium on Algorithm Engineering and Experiments, ALENEX 2023, 2023. URL: https://doi.org/10.1137/1.9781611977561.CH3.
  4. Hideto Asashiba, Mickael Buchet, Emerson G. Escolar, Ken Nakashima, and Michio Yoshiwaki. On interval decomposability of 2D persistence modules. Computational Geometry, 105-106:101879, 2022. URL: https://doi.org/10.1016/j.comgeo.2022.101879.
  5. Ulrich Bauer, Michael Kerber, Fabian Roll, and Alexander Rolle. A unified view on the functorial nerve theorem and its variations. Expo. Math., 41(4):125503, 2023. URL: https://doi.org/10.1016/j.exmath.2023.04.005.
  6. Ulrich Bauer and Luis Scoccola. Generic multi-parameter persistence modules are nearly indecomposable, 2023. URL: https://arxiv.org/abs/2211.15306.
  7. Håvard Bakke Bjerkevik. Stabilizing decomposition of multiparameter persistence modules, 2023. URL: https://arxiv.org/abs/2305.15550.
  8. Andrew J. Blumberg and Michael Lesnick. Stability of 2-parameter persistent homology. Foundations of Computational Mathematics, 2022. URL: https://doi.org/10.1007/s10208-022-09576-6.
  9. Omer Bobrowski and Robert J. Adler. Distance functions, critical points, and the topology of random Čech complexes. Homology, Homotopy and Applications, 16(2):311-344, 2014. URL: https://doi.org/10.4310/hha.2014.v16.n2.a18.
  10. Omer Bobrowski and Matthew Kahle. Topology of random geometric complexes: a survey. Journal of Applied and Computational Topology, 1(3-4):331-364, 2018. URL: https://doi.org/10.1007/s41468-017-0010-0.
  11. Omer Bobrowski, Matthew Kahle, and Primoz Skraba. Maximally persistent cycles in random geometric complexes. The Annals of Applied Probability, pages 2032-2060, 2017. URL: https://doi.org/10.1214/16-AAP1232.
  12. Omer Bobrowski and Primoz Skraba. A universal null-distribution for topological data analysis. Scientific Reports, 13(1), July 2023. URL: https://doi.org/10.1038/s41598-023-37842-2.
  13. Magnus Bakke Botnan and Christian Hirsch. On the consistency and asymptotic normality of multiparameter persistent Betti numbers. Journal of Applied and Computational Topology, December 2022. URL: https://doi.org/10.1007/s41468-022-00110-9.
  14. Magnus Bakke Botnan and Michael Lesnick. An introduction to multiparameter persistence. In 2020 International Conference on Representations of Algebras, pages 77-150. EMS Press, November 2023. URL: https://doi.org/10.4171/ecr/19/4.
  15. Mickaël Buchet, Bianca B. Dornelas, and Michael Kerber. Sparse higher order čech filtrations. In 39th International Symposium on Computational Geometry, SoCG 2023, June 12-15, 2023, Dallas, Texas, USA, pages 20:1-20:17, 2023. URL: https://doi.org/10.4230/LIPICS.SOCG.2023.20.
  16. Mickaël Buchet and Emerson G. Escolar. Realizations of indecomposable persistence modules of arbitrarily large dimensions. Journal of Computational Geometry, 13(1), 2022. URL: https://doi.org/10.20382/jocg.v13i1a12.
  17. Gunnar Carlsson and Afra Zomorodian. The theory of multidimensional persistence. Discrete Comput. Geom., 42(1):71-93, 2009. URL: https://doi.org/10.1007/s00454-009-9176-0.
  18. René Corbet, Michael Kerber, Michael Lesnick, and Georg Osang. Computing the multicover bifiltration. In 37th International Symposium on Computational Geometry, SoCG 2021, pages 27:1-27:17, 2021. URL: https://doi.org/10.4230/LIPICS.SOCG.2021.27.
  19. Tamal K. Dey and Cheng Xin. Computing bottleneck distance for 2-D interval decomposable modules. In 34th International Symposium on Computational Geometry, SoCG 2018, pages 32:1-32:15, 2018. URL: https://doi.org/10.4230/LIPIcs.SoCG.2018.32.
  20. Tamal K. Dey and Cheng Xin. Generalized persistence algorithm for decomposing multiparameter persistence modules. Journal of Applied and Computational Topology, 6:271-322, 2022. URL: https://doi.org/10.1007/s41468-022-00087-5.
  21. Herbert Federer. Curvature measures. Transactions of the American Mathematical Society, 93(3):418-491, 1959. URL: https://doi.org/10.2307/1993504.
  22. Miroslav Fiedler. Matrices and graphs in geometry, volume 139 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 2011. URL: https://doi.org/10.1017/CBO9780511973611.
  23. Ulderico Fugacci, Michael Kerber, and Alexander Rolle. Compression for 2-parameter persistent homology. Computational Geometry, 109:101940, 2023. URL: https://doi.org/10.1016/j.comgeo.2022.101940.
  24. Yasuaki Hiraoka, Ken Nakashima, Ippei Obayashi, and Chenguang Xu. Refinement of interval approximations for fully commutative quivers, 2023. URL: https://arxiv.org/abs/2310.03649.
  25. Yasuaki Hiraoka, Tomoyuki Shirai, and Khanh Duy Trinh. Limit theorems for persistence diagrams. Ann. Appl. Probab., 28(5):2740-2780, 2018. URL: https://doi.org/10.1214/17-AAP1371.
  26. Matthew Kahle. Random geometric complexes. Discrete Comput. Geom., 45(3):553-573, 2011. URL: https://doi.org/10.1007/s00454-010-9319-3.
  27. Günter Last and Mathew Penrose. Lectures on the Poisson process, volume 7. Cambridge University Press, 2017. Google Scholar
  28. Michael Lesnick and Matthew Wright. Interactive Visualization of 2-D Persistence Modules. URL: https://arxiv.org/abs/1512.00180.
  29. Michael Lesnick and Matthew Wright. Computing minimal presentations and bigraded betti numbers of 2-parameter persistent homology. SIAM Journal on Applied Algebra and Geometry, 6(2):267-298, 2022. URL: https://doi.org/10.1137/20M1388425.
  30. A. W. Marshall, D. W. Walkup, and R. J.-B. Wets. Order-preserving functions: Applications to majorization and order statistics. Pacific J. Math., 23:569-584, 1967. Google Scholar
  31. R. A. Parker. The computer calculation of modular characters (the meat-axe). In Michael D. Atkinson, editor, Computational Group Theory, pages 267-274. New York: Academic Press, 1984. Google Scholar
  32. Bernard W. Silverman. Density Estimation for Statistics and Data Analysis. Springer Netherlands, 1986. URL: https://doi.org/10.1007/978-1-4899-3324-9.
  33. Larry Wasserman. All of statistics: a concise course in statistical inference, volume 26. Springer, 2004. Google Scholar
  34. D. Yogeshwaran and Robert J. Adler. On the topology of random complexes built over stationary point processes. Ann. Appl. Probab., 25(6):3338-3380, 2015. URL: https://doi.org/10.1214/14-AAP1075.
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